This invention relates to a numerical method for solving the Navier-Strokes equation and has application in computer simulation of viscous, incompressible laminar fluid flow; in particular, it relates to a method using finite element-control volume techniques, wherein the velocity and pressure of the fluid are evaluated at the same order to preserve accuracy of solution without sacrificing computation efficiency.
Modeling incompressible, viscous flow generally involves solving the Navier-Stokes equation to obtain the velocity field. The Navier-Stokes equation describes the fundamental law of mass conservation and momentum balance of incompressible viscous flow. For a given pressure field, there is no particular difficulty in solving the momentum equation. By substituting the correct pressure field into the momentum equation, the resulting velocity field satisfies the continuity constraint. The difficulties in calculating the velocity field lies when the pressure field is unknown. The pressure field is indirectly specified via the continuity equation. Yet, there is no obvious equation for obtaining pressure.
The difficulty associated with the determination of pressure has led to methods that eliminate pressure from the momentum equations. For example, in the well-known xe2x80x9cstream-function/vorticity methodxe2x80x9d described by others and Gosman et. al. (In Heat and Mass Transfer in Recirculating Flows, Academic, New York, 1969.) The stream-function/vorticity method has some attractive features, but the pressure, which has been so cleverly eliminated, frequently happens to be a desired result or even an intermediate outcome required for the calculation of density and other fluid properties. In addition, the method cannot be easily extended to three-dimensional situations. Because most practical problems are three-dimensional, a method that is intrinsically restricted to two dimensions suffers from a serious limitation.
Numerical methods have been developed for solving the momentum equation and obtaining the pressure field. The SIMPLE (Semi-Implicit Method for Pressure-Linked Equations by Patenkar, in Numerical Heat Transfer and Fluid Flow, McGraw Hill, 1980), and SIMPLER (SIMPLE Revised) algorithms are well-known examples of the numerical methods. The SIMPLE and SIMPLER methods discretize the flow domain, and transform the momentum equation into a set of algebraic equations. The pressure-gradient term is represented in the discretized momentum equations by the pressure drop between two grid points. One consequence of this representation is the non-uniqueness of the solution of the pressure field. As long as the pressure values at alternate grid points are uniform, the discretized form of the momentum equations is incapable of distinguishing a checker-board pressure field from a uniform pressure field. However, the potential to result in a checker-board pressure field should make the solution unacceptable. Another consequence of this representation is that pressure is, in effect, taken from a coarser grid than the one actually employed. This practice diminishes the accuracy of the solution. It has been recognized that the discretizing equation does not demand that all the variables be calculated on the same grid points. The SIMPLE and SIMPLER methods resolve the checker boarding of pressure by using a xe2x80x9cstaggeredxe2x80x9d grid for the velocity component. In the staggered grid, the velocity components are calculated on a grid staggered in relation to the normal control volume around the main grid points. A computer program based on the staggered grid must carry all the indexing and geometric information about the locations of the velocity components and must perform interpolation for values between grid points. Furthermore, a xe2x80x9cstaggeredxe2x80x9d grid is difficult to apply to a flow domain that is meshed into irregularly shaped finite elements.
Reduced order and reduced intergration methods have been suggested for modeling fluid flow in complex fluid domains. The reduced order method suggests that pressure be evaluated in a coarser grid, e.g. at alternate nodal positions of the finite element mesh (Baliga and Patankar, A Contol Volume Finite-Element Method for Two Dimensional Fluid Flows and Heat Transfer, in Nutmerical Heat Transfer, 6, 245-261 (1980)). The reduced order method diminishes the accuracy of the pressure field, and therefore compromises the accuracy of the subsequent mass flux calculations. Additionally, the method has been shown to be inadequate for modeling a fluid flow with a moving free surface. The reduced integration method suggests that less than full integration should be used for the pressure term. This method has shown some success in specific applications, but there are many limitations to its general application. Furthermore, reduced integration methods also sacrifice accuracy of the solution.
There is therefore a need for a method to solve the momentum equations of incompressive, viscous fluid flows such that accuracy of solution is preserved without sacrificing computation efficiency.
Accordingly, one preferred object of the present invention is to provide a numerical method for solving the Navier-Stokes equation of viscous incompressible, laminar fluid flow and wherein the velocity and pressure of the fluid are evaluated at the same order to preserve accuracy of solution without sacrificing computation efficiency.
Another preferred object of the present invention is to provide a numerical method for solving the momentum equation and providing a solution of such accuracy that the global mass conservation can be maintained to machine round-off tolerances.
One embodiment of the present invention is a numerical method for solving the coupled momentum and mass conservartion equation. The method resides and operates within a computer processor. The method defines a gauss point velocity vector as the average of the nodal velocity vectors. In essense, the expression of gauss point velocity vector is in the same order as the pressure gradient over each element. Because the gauss point fluxing vector field is xe2x80x9ccenteredxe2x80x9d inside the pressure field defined at the nodes, no checker boarding will occur. Additionally, because the gauss point velocities and pressure fields are evaluated from the same number of nodal positions, the accuracy of the solution is preserved. The predicted gauss point fluxing velocity field conserves mass to machine round-off levels for element types in which the pressure gradient and velocities can be evaluated on a consistent basis. In two dimensions, 3 noded, linear triangular elements will conserve mass to machine round-off tolerances. In three dimensions, 4 noded, linear tetrahedral elements will conserve mass to machine round-off tolerances.
One aspect of the present invention is to provide a numerical method operating within a processor environment for solving the momentum equation in a computer simulation of a viscous, incompressible, laminar flow within a flow domain comprising: (a) discretizing the flow domain into a mesh of elements having nodes and fluxing surfaces; (b) forming momentum equations; (c) computing nodal hat velocities at each node; (d) calculating gauss point hat velocities and gauss point coefficient, wherein the gauss point hat velocities and gauss point coefficient are average value of nodal hat velocities and nodal coefficient; (e) updating velocity field; and (f) checking if velocity field has converged, and returning to the forming momentum equation if the velocity field has not converged.
Other objects, embodiments, forms, aspects, features, advantages, and benefits will become apparent from the description and drawings provided herein.